## Topology and New Kinds of Order in Condensed Matter Physics (PH2246)

**Lecturer: **Frank Pollmann

**Teaching Assistants:** Julian Bibo, Shenghsuan Lin, Nico Kirchner

**Lectures:** Monday 10:15 - 11:45; Wednesday 10:15 - 11:45 in Lecture Hall 5901.EG.051 (EI Building)

**Tutorials:** Friday 10:00 - 12:00

With the discovery of the integer and fractional quantum Hall effect in the 1980s, it was realized that not all phases of matter occurring in nature can be understood using Landau’s theory. The quantum Hall state represents a distinct phase of matter which can occur even when there is no local order parameter or spontaneous breaking of a global symmetry. Phases of this new kind are now usually referred to as topological phases. This lecture course gives an introduction to different theoretical aspects of topological phases and their experimental signatures. The following topics are covered in the course

The course will be offered this semester at the announced times and will be based on web conferencing. Further details will be sent by email to all participants registered in campus online. Please sign up online to be able to receive the latest information. The course material including lecture notes and excercises will be posted on moodle.

**Outline:**

**Introduction**

(1) Landau Theory, Kosterlitz Thouless transition

(2) Quantum Phases of matter, second quantization, quantum-Ising model

**Topological Band structures**

(3) Su-Schrieffer-Heeger Model: Edge modes, topological invariant

(4) Berry phase and Chernnumber

(5) Haldane model

(6) 2D TR-invariant topological insulators

(7) 3D TR-invariant topological insulators

(8) 3D Weyl semimetals

(9) Topological superconductors: Kitaev chain

(10) Periodic table of topological band structures

**Quantum Hall Effect**

(11) Phenomenology, Disorder

(12) Laughlin’s argument

(13) Thouless-Kohmoto-Nightingale-den Nijs (TKNN)

**Topological order**

(14) Fractional quantum Hall effect

(15) Quantum spin systems and entanglement

(16) Toric code and anyons

(17) A glimpse at topological quantum computing

(18) Quantum Spin Liquids

**Interacting Symmetry Protected Topological Phases **

(19) Symmetry Protected Topological Phases in 1D

(20) Symmetry Protected Topological Phases in 2D

**Fractionalization and emergent electrodynamics **

(21) 3D Spin ice and monopoles

**Literature:**

Quantum Field Theory of Many-Body Systems, X.-G. Wen, Oxford.

Graduate Text Lecture notes by John Preskill: http://www.theory.caltech.edu/~preskill/ph219/topological.pdf.

Lecture notes by Alexei Kitaev and Chris Laumann: http://arxiv.org/pdf/0904.2771v1.pdf

Review article by Chetan Nayak et al.: http://arxiv.org/pdf/0707.1889v2.pdf.